-If you're running Universal Suffrage with a governor automating the city they'll rush buildings every so often when you are raking in a huge gold surplus in order to quickly improve those cities. Having a large gold surplus per turn or hefty bank (it seems to be some combination of both) triggers this as long as they still have improvement buildings to construct.
-If you turn your science slider all the way down, or cripple your economy entirely, they will build research/wealth respectively, often to the detriment of relieving the problem instead by actually building something useful like Markets or Courthouses.
-On the subject of relief, I'm not sure if they whip anything but they definitely never do it in order manage happiness, and with citizen automation they'll even avoid growing too big on purpose. I find it more useful to overgrow cities if I am too busy to manage them and I'm still in Slavery; the extra population is free production and the right amount population shaved on a big enough project both eliminates the unhappiness and doesn't waste the hammers, for the least
possible.
-They don't read specialist economy strats very well and think you are crippling your city by running too many. This leads to them tending to make odd decisions when picking tiles to work or picking a new building (ex.: low on production in a GP farm? Let's build a quite expensive Forge at 1 hammer per turn! >_>)
-Specialist assignment works the same if you turn citizen automation on; run your EP slider at 0% with good science/gold standing and a lack of courthouses/jails/what-have-you, and every time you get a new citizen to work a fully staffed city or capture an enemy one (where citizen automation is on by default after you take it), they'll run spy specialists, often tons of them. This sucks because they don't usually change their minds later, screwing up GP generation if you don't watch them.
-They build culture in newly founded/captured cities if you have Music, run artists if you have Caste System active, or build Monuments/Theaters in order to get the first border pop.